A259933 Amicable pairs (x < y) ordered by nondecreasing sum (x + y) and then by increasing x.

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 66928, 66992, 67095, 71145, 63020, 76084, 69615, 87633, 79750, 88730, 100485, 124155, 122368, 123152, 122265, 139815, 141664, 153176, 142310, 168730, 171856, 176336, 176272, 180848, 185368, 203432, 196724, 202444, 280540, 365084, 308620, 389924

Offset: 1

Author

Omar E. Pol, Jul 09 2015

Keywords

• nonn

Comments

A pair of numbers x and y is called amicable if the sum of the proper divisors (or aliquot parts) of either one is equal to the other.

By definition a property of the amicable pair (x, y) is that x + y = sigma(x) = sigma(y).

The amicable pairs (x < y) are adjacent to each other in the list.

Also A260086 and A260087 interleaved.

Another version (A259180) lists the amicable pairs (x < y) ordered by increasing x.

Amicable numbers A063990 are the terms of this sequence in increasing order.

First differs from both A063990 and A259180 at a(17).

Examples

			-----------------------------------
       Amicable pair         Sum
          x      y          x + y
-----------------------------------
n     A260086 A260087      A259953
-----------------------------------
1        220     284          504
2       1184    1210         2394
3       2620    2924         5544
4       5020    5564        10584
5       6232    6368        12600
6      10744   10856        21600
7      12285   14595        26880
8      17296   18416        35712
9      66928   66992       133920
10     67095   71145       138240
11     63020   76084       139104
12     69615   87633       157248
...      ...     ...          ...
32    609928  686072      1296000
33    643336  652664      1296000
...
The sum of the proper divisors (or aliquot parts) of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. On the other hand the sum of the proper divisors (or aliquot parts) of 284 is 1 + 2 + 4 + 71 + 142 = 220. Note that 220 + 284 = sigma(220) = sigma(284) = 504. The sum 220 + 284 = 504 is the smallest sum of an amicable pair, so a(1) = 220 and a(2) = 284.
Note that some pairs (x, y) share the same sum (x + y), for example: (609928 + 686072) = (643336 + 652664) = sigma(609928) = sigma(686072) = sigma(643336) = sigma(652664) = 1296000, thus in the list first appears the pair (609928, 686072) and then (643336, 652664) because 609928 < 643336.
		

Links

• Laszlo Hars, Performance compared to mathematica Julia-users (2014)
• Khelleos, Amicable numbers, CyberForum.ru (2011)
• OEIS Wiki, Amicable numbers (This page needs work)
• Wikipédia, Barátságos számok (contains a mistake: A063990 should be replaced with A259933)

Crossrefs

Cf. A000203, A063990, A259180, A259953, A260086, A260087.

Formula

a(2n-1) + a(2n) = A000203(a(2n-1)) = A000203(a(2n)) = A259953(n).